3 width sides (of width x), and 2 length sides (of length y). First add up all the lengths and widths and sum to 1000 feet:
Perimeter:
3x + 2y = 1000 2y = 1000 - 3x y = 500 - 1.5x
Area:
A(x) = xy
Combine equations:
A(x) = x(500 - 1.5x) = 500x - 1.5x^2
Since A(x) = x(500 - 1.5x), we want \(x\geqslant 0, 500 - 1.5x \geqslant 0\), so \(0\leq x\leq 333.33\).
Set A’(x) = 0,
A’(x) = 500 - 3x = 0, x = 500/3
A(500/3) = 500/3(500 - 1.5(500/3)) = 41666.67 feet
We set y = 500 - 1.5x, y = 250,
3 side of length is 500/3, and 2 side of length is 250, total area is 41666.67 feet square.